How are wholesale electricity prices set?

The grid operator asks for offers to sell electricity for the next, say, 5 minutes or half an hour, to fill expected electricity demand over that period.  It accepts the cheapest offers first, then the next cheapest, and so on, until the total of all the offers to supply matches expected demand.  The price of the last supplier in that stack, the most expensive, is then paid to all suppliers.  This is called merit order pricing.  

Wind and solar have nearly zero marginal cost, because they have no fuel costs, and having built a wind or solar farm, its owners might as well use it as long as the price it receives is above zero.  So they offer first, at a low prices, because they know that the price they'll actually get will be higher.  Next in the queue comes nuclear.  Nuclear can't turn itself off and on at 5-minute or 30-minute intervals, so it is important that it sells its output.  Its marginal (as opposed to total) cost is lower than coal or gas, because of fuel and maintenance, so it offers into the stack at its marginal cost, certain that it will get at least that when the final price is settled.

Next comes brown coal (lignite).  Marginal/operating cost is higher than nuclear but lower than black (hard) coal, but like nuclear, coal power stations can't be rapidly scaled up and down.  So they'd make a middling offer, hoping that the final price will be higher.  And so it goes, with black coal more costly than brown,  gas (outside the US) more expensive than coal, and oil more expensive still.  Remember, each generator has a slightly different marginal cost, depending on its fuel supply contracts.

Some gas generators, called peaking plants, can rapidly scale up supply, so they tend to put in high offers.  If the merit order auction doesn't reach that price for that period, no matter, they'll make money when it does, which happens when renewable supply is very low or demand is very high.   

The implication of this process is that  as output of renewables rises, it pushes the highest marginal cost generators off the table, driving down wholesale prices.  Actually, it's worse than that.  Since total costs are higher than marginal costs, but marginal costs of renewables are zero, fossil fuel generators, except peaking gas plants, might not get high enough prices on average to cover total costs, leading eventually to bankruptcy.  For example,  lots of solar at midday drives down wholesale prices, as does lots of wind at late at night when demand is low.  Wholesale prices can even go negative, because coal power stations can't scale down output to zero.  If these periods of low/negative prices go on long enough, the very viability of coal generators is threatened, even if there are times when strong demand drives up wholesale prices.  The obvious solution for fossil fuel generators is to install batteries so that, for example,  at the mid-day solar peak, shunt their output into the batteries, to be released at the late afternoon/evening demand peak. 

The chart below, from Clean Energy Wire, shows how the merit order effect works.

Power demand is the vertical dashed line, price is the horizontal one. 
As output of wind/solar increases, the output cost curves shifts to the right.
The wholesale prices of electricity drops as a result.